Stephen Kovacevich was born in San Pedro, Los Angeles, California, to a Croatian father and an American mother. When his mother remarried, his name was changed to Stephen Bishop, the name under which he performed in his early career. He later discovered that he was often being confused with the singer and guitarist Stephen Bishop. To avoid the confusion, he began performing as Stephen Bishop-Kovacevich and later simply as Stephen Kovacevich.[2]

From 1961 to 1967 he was married to novelist and psychotherapist Bernardine Bishop née Wall, who died in 2013. They had two sons, Matt Bishop (a director of the McLarenFormula 1 team) and Foff Bishop (a West Sussex fireman).[4]

In 1967, he made his New York debut and since then he has toured Europe, the United States, the Far East, Australia, New Zealand and South America.

As a soloist and conductor, he has frequently performed and recorded works by, amongst others, Mozart, Beethoven, Schubert, Brahms and Bartók.